The sum of powers of subtree sizes for conditioned Galton–Watson trees

Abstract

We study the additive functional Xn(α) on conditioned Galton–Watson trees given, for arbitrary complex α, by summing the αth power of all subtree sizes. Allowing complex α is advantageous, even for the study of real α, since it allows us to use powerful results from the theory of analytic functions in the proofs. For Reα<0, we prove that Xn(α), suitably normalized, has a complex normal limiting distribution; moreover, as processes in α, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence extensions of, limiting distribution results for α in various regions of the complex plane. We focus mainly on the case where Reα>0, for which Xn(α), suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution ξ of the conditioned Galton–Watson tree, assuming only that Eξ=1 and 0<Varξ<∞. Under a weak extra moment assumption on ξ, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when Reα>12, the limit random variable Y(α) can be expressed as a function of a normalized Brownian excursion.